Set mode passive location in TOA/TDOA modes

ABSTRACT

The present invention addresses the resolving of the problems associated with the passive location of targets in TOA (Time of Arrival) or TDOA (Time Difference of Arrivals) mode. The method of passively locating a target in TOA or TDOA mode implements a meshing (subdivision) into blocks of the space in which the location area is situated. The set of the blocks that form this mesh is analyzed iteratively. On each iteration, each block of interest is subdivided into smaller identical subblocks. A block of interest is, according to the invention, a block including at least one point belonging to the location area being sought for which the shape is to be determined. The iterative process is stopped when the size of the subblocks obtained on the current iteration corresponds to the desired resolution. The invention applies in particular to the 2D or 3D location systems that include TOA and TDOA modes or mixed modes.

CROSS-REFERENCE TO PRIOR APPLICATION

This is a U.S. National Phase Application under 35 U.S.C. §371 ofInternational Application no. PCT/EP2007/063910, filed Dec. 13, 2007,and claims benefit of French Patent Application No. 06/10961, filed Dec.15, 2006, both of which are incorporated herein. The InternationalApplication was published in French on Jun. 19, 2008 as WO 2008/071777under PCT Article 21 (2).

FIELD OF THE INVENTION

The present invention addresses the resolving of the problems associatedwith the passive location of targets in TOA (Time Of Arrival) or TDOA(Time Difference Of Arrival) mode.

For TOA, the invention makes it possible to locate any target byanalyzing the arrival times, on one and the same receiver, of the wavestransmitted by one or more transmitters and reflected by the target.

For TDOA, the invention makes it possible to locate a transmittingtarget by measuring differences of arrival time of the transmitted waveon a number of receivers that are synchronized and scattered in space.

The present invention relates more particularly (but not exclusively) tothe field of passive radars.

By its generic nature, the present invention also addresses all the modecombinations (multiple TOA, multiple TDOA, or even mixed TOA/TDOA).

CONTEXT OF THE INVENTION—PRIOR ART

The basic principle of the passive location methods, whether in TOA orTDOA mode, is to determine the positioning of targets by using theinformation supplied simultaneously by different information sources.

In TOA mode, or “Time Of Arrival” mode, interest is focused on thesignals transmitted by one or more transmitters and the same signalsreflected by a target. The measurement, for a given transmitter, of thedelay between the forward path (transmitter→receiver) and the reflectedpath (transmitter→target→receiver) is used to define a location curve(or a surface if the problem is dealt with in 3D) that takes the form ofan ellipse or an ellipsoid.

Thus, if a number of transmitters are analyzed simultaneously, andprovided that there is the capability to receive the forward andreflected paths, it is possible to determine the position of the targetthat is sought by determining the mutual points/areas of intersectionsof the different location curves/surfaces.

In TDOA mode, or “Time Difference Of Arrival” mode, interest is focusedon the location of transmitting targets by means of a number ofsynchronized receivers (at least two), a main receiver and one (or more)secondary receivers. As in the TOA mode, the position of thetransmitting target is then determined geometrically from locationcurves (hyperbolas or hyperboloids) established from the measurements ofthe delays between the different signals originating from the target andarriving at different instants on each of the receivers.

One of the problems raised by passive location stems from thetaking-into-account of the uncertainty of the measurement delivered bythe receivers. In practice, to produce an accurate location of thetarget, it is necessary to know as accurately as possible the locationarea that is compatible with the uncertainties affecting themeasurements. Consequently, the location curves are in reality locationareas, each area being situated between two extreme curves, the spacingof which depends on the accuracy of the receivers. That way, the pointsof intersections of the different location curves ideally obtained byusing a number of transmitting sources (TOA mode) or a number ofreceivers (TDOA mode) are replaced in practice by areas of intersectionwithin which these points are situated.

Then, the search for the location areas that are compatible with themeasurements firstly involves the mathematical characterization of saidareas followed by the search for them in a space of interest (i.e. thespace in which the presence of a target is sought).

The usual methods used to perform this search are generally grid methodswhich involve finely meshing all the space in which the targets aresought, that is, a space that is vast enough to contain the area ofuncertainty and systematically analyzing each mesh to check whether itbelongs to the location area. There are also algebraic methods of theleast-square type, or even probabilistic methods, the complexity (andtherefore the complexity of implementation) of which increases with thenumber of information sources. Apart from the grid methods, none ofthese methods provides a way of finely restoring the areas ofuncertainties associated with the measurement errors (they give onlyerror ellipses or ellipsoids). In addition, the grid methods require alarge computation capability to process each mesh sufficiently quicklyand determine the location area sufficiently quickly.

DESCRIPTION OF THE INVENTION

One aim of the invention is to benefit from the advantages of the gridprocessing operations notably in terms of resolution, yet withoutsuffering the drawbacks thereof.

To this end, the subject of the invention is a method of passivelylocating a target in TOA or TDOA mode that implements a successivesubdivision into blocks of an initial space (in which a target is to belocated). The set of blocks is analyzed iteratively. On each iteration,each block of interest is subdivided into smaller identical subblocks. Ablock of interest is, according to one or more embodiments of theinvention, a block in which at least one point belongs to the locationarea being sought. The iterative process is stopped when the size(resolution) of the subblocks obtained on the current iterationcorresponds to the desired resolution.

The set of blocks resulting from this process provides an approximationof the shape of the location area that is sought (if the latter existsin the initial space).

More specifically, its subject is a method of passively locating atarget in TOA or TDOA mode implementing a successive subdivision intoblocks of an initial space (in which a target is to be located) and asearch within each block for the presence of points belonging to thelocation area being sought. This invention is also characterized by thefact that the subdivision and the search are performed in the form ofiterative steps on a selection of candidates blocks, modified on eachiteration, so that, on each iteration, the blocks of the selectionobtained on completion of the preceding iteration are searched to findthe blocks containing at least one point belonging to the location areabeing sought. The blocks not containing any point are subsequentlyexcluded from the selection, whereas the blocks containing at least onepoint are re-subdivided into subblocks and replaced in the selection bythe duly formed subblocks. The selection obtained on each iterationdefines the location area with a resolution that increases on eachiteration.

According to one or more embodiments of the invention, the iterationsare stopped when the subblocks forming the selection define the locationarea with the desired resolution.

The method according to one or more embodiments of the invention alsoincludes an initialization step in which a first block [X₀] is definedthat corresponds to an “a priori” search space, this block constitutingthe initial selection.

According to a preferred embodiment, the method according to theinvention mainly includes:

-   -   an initialization step in which a first block [X₀] is defined        that corresponds to an a priori search space and that leads to        the formation of an initial list L₀ ⁰ formed from the block        [X₀],    -   an iterative step including:        -   subdividing each block [X_(n)] of the current list L₀ ^(i-1)            into four adjoining subblocks [X_(n) ¹], [X_(n) ²], [X_(n)            ³] and [X_(n) ⁴],        -   searching for and selecting from the duly constituted            subblocks those containing at least one point forming part            of the location area,        -   updating the current list to obtain a list L₀ ^(i) in which:            -   a) the blocks [X_(n)] for which no subblock has been                selected are deleted,            -   b) the other blocks are replaced by the selected                subblock [X_(n) ^(j)],                the iterative step also includes an operation to compare                the size Δ^(L) ^(i) ⁰ ⁽¹⁾ of the blocks constituting the                list L₀ ^(i), the method being stopped when the size                Δ^(L) ^(i) ⁰ ⁽¹⁾ is greater than the desired resolution                objective Δ^(obj).

According to this embodiment, a block [X_(n) ^(j)] is selected if atleast one of its points satisfies the criterion defined as follows:

${{{- 0} \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack}{\lbrack b\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}}},{{{- 0} \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}}},{{{- 0} \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = {\quad{{\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} + \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}},{{{- 0} \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = {\quad{{\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}},}}}}}}$

The invention relies on the set mode approach used to locate the target.The measurements deriving from the various sensors are modeled byintervals (i.e., they include a bounded error). Knowing thesemeasurements, one or more embodiments of the invention uses an iterativeprocess based on an ad hoc set mode criterion (i.e. dependent on theproblem, TOA/TDOA single/multiple sensors) to find and approach with thedesired resolution, all the areas of the space that are likely, in lightof the measurements, to contain a target. The location is set mode inthe sense that one or more embodiments of the invention provides asolution set to the location problem (i.e., a set of target positionsguaranteed to contain the true position of the target).

Unlike a grid method, there is no need to mesh all the space, theiterative process used in one or more embodiments of the invention makesit possible to concentrate directly on the regions of interest.

DESCRIPTION OF THE FIGURES

The features and benefits of the invention will be better appreciatedfrom the description that follows, which explains the invention througha particular embodiment taken as a nonlimiting example and based on theappended figures, which represent:

FIG. 1, a typical single-transmitter geometrical configuration forimplementing the TOA mode,

FIG. 2, a representation of the set of location curves ideally obtainedin TOA mode with three transmitting sources,

FIG. 3, a typical two-receiver geometrical configuration forimplementing the TDOA mode,

FIG. 4, a representation of the set of location curves ideally obtainedin TDOA mode with three secondary receivers,

FIG. 5, the representation of a location area actually obtained in TOAmode with one transmitting source,

FIG. 6, the representation of a location area actually obtained in TDOAmode with one receiver,

FIG. 7, a theoretical flow diagram of the method according to theinvention,

FIG. 8, an illustration of the application of the method according tothe invention to the TOA 2D mode with a single-transmitter system,

FIG. 9, an illustration of the application of the method according tothe invention to the TDOA 2D mode with a single secondary receiversystem,

FIG. 10, an illustration of the application of the method according theinvention to the TOA 2D mode with a two-transmitter system, and

FIG. 11, an illustration of the application of the method according tothe invention to the TDOA 2D mode with a two secondary receivers system.

DETAILED DESCRIPTION

Interest is focused initially on FIG. 1 which schematically shows theideal operating principle of the TOA location mode. In the interest ofclarity of the explanation, the model illustrated here is atwo-dimensional model corresponding to the analysis of the signalsobtained from a single transmitter (2D single-transmitter model).

As the figure illustrates, this “single transmitter” operating modeinvolves a transmitting source 12 (transmitter), a receiver 11, and atarget 13 that is to be located. To locate the target, the receiver 11measures the time delay that exists between the received waveoriginating directly from the transmitter 12 (forward wave), and thereceived wave originating from the reflection of the wave transmitted bythe source 12, on the target 13 that is to be located (reflected wave).

In such a configuration, if the distance between the transmitting source12 and the receiver 11 is denoted L, the distance between the source 12and the target 13 R_(T) and the distance between the target and thereceiver R_(R), the location of the target 13 involves measuring thedelay between the forward and reflected waves and determining from thisdelay, by any known appropriate method, the distance R_(b) traveled bythe reflected wave, defined by:R _(b) =R _(T) +R _(R)The target 13 is then located as illustrated by the curve 21 of FIG. 2on an ambiguity ellipse, the foci 22 and 23 of which are the position ofthe transmitting source 12 and that of the receiver 11, and the majoraxis (semi-major axis) of which has the length R_(b)/2.

Consequently if a Cartesian frame of reference xOy is defined that iscentered on the middle of the line segment [R_(x), T_(x)] 14 linking thesource 12 and the receiver 11, and the vector {right arrow over (Ox)} ofwhich is collinear to the vector {right arrow over (R_(x)T_(x))}, theequation of this location ellipse 21 in the frame of reference xOy isexpressed by the following equation:

$\begin{matrix}{{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}} = 1} & \lbrack 1\rbrack\end{matrix}$with:a=R _(b)/2and b=√{square root over (a ² −L ²/4)}To refine the location of the target 13, it is obviously necessary tohave a number of transmission sources. That way, for one and the sametarget 13, a location ellipse can be associated with each source, theintersections of these ellipses defining the possible positions of thesource to be located. The curves 21 and 24 of FIG. 2 illustrate theresults obtained with a “2D two-transmitter” configuration from whichtwo location ellipses are obtained, the foci 22, 23 and 25 of which arerespectively the receiver and the first source for the ellipse 21, andthe receiver and the second source for the ellipse 24. The set of thepossible places of location of the target 13 then includes the twopoints of intersection 26 and 27 of the two curves.

Consequently, to determine the position of the target 13 withoutambiguity, it is necessary to have at least one additional transmittingsource (“2D multiple-transmitter” configuration), the intersection ofthe three location ellipses 21, 24 and 28 defining a single commonpoint, the point 26 for example, on which the target 13 is situated.

Interest is then focused on FIG. 3 which schematically shows the idealoperating principle of the TDOA location mode. In the interest ofclarity of the explanation, the model illustrated here is, as for theTOA mode presentation, a two-dimensional model corresponding to theanalysis of the signals obtained from a transmitting target 31 andreceived by two receivers 32 and 33 distant from each other, a referencereceiver, called main receiver, and a so-called secondary receiver,synchronized on the reference receiver (2D single secondary receivermodel).

As illustrated by FIG. 3, this “secondary single-receiver” operatingmode involves a transmitting target 31 (transmitter) that is to belocated, a reference receiver 32, and a secondary receiver 33. To locatethe target, the time delay that exists between the instant of receptionby the reference receiver of the wave transmitted by the target 31 andthe instant of reception of this same wave by the secondary receiver 33is then analyzed.

In such a configuration, if the distance between the two receivers 32and 33 is denoted L, the distance between the transmitting target 31 andreference receiver 32 is denoted R_(R) and the distance between thetransmitting target 31 and the secondary receiver 33 is denoted R_(R1),the location of the target 31 includes measuring the delay between thewaves received by the two receivers and determining from this delay, byany known appropriate method, the difference in distance traveled R_(d1)defined by:R _(d1) =R _(R1) −R _(R)

The target 31 is then located as illustrated by the curve 41 of FIG. 4on a hyperbola having foci 42 and 43 which are the position of thereference receiver and that of the secondary receiver and for which thedistance between peaks has the value R_(d1)/2.

Consequently, if a Cartesian frame of reference xOy is defined that iscentered on the middle of the line segment [R_(x), R_(x1)] 34 linkingthe reference receiver 32 and the secondary receiver 33, and the vector{right arrow over (Ox)} of which is collinear to the vector {right arrowover (R_(x)R_(x1))}, the equation for this location hyperbola 41 in theframe of reference xOy is expressed:

$\begin{matrix}{{\frac{x^{2}}{c^{2}} - \frac{y^{2}}{d^{2}}} = 1} & \lbrack 2\rbrack\end{matrix}$with:c=R _(d1)/2and d=√{square root over (L ²/4−c ²)}

To refine the location of the target 31, it is obviously necessary tohave a number of secondary receivers. That way, for one and the sametarget 31, a location hyperbola can be associated with each referencereceiver/secondary receiver pairing. The intersections of thesehyperbolas then define the possible positions of the target. The curves41 and 44 of FIG. 4 illustrate the results obtained with a “2D twosecondary receivers” configuration from which two location hyperbola areobtained, the foci 42, 43 and 45 of which are respectively the referencereceiver R_(x) and the secondary receiver R_(x1) for the hyperbola 41and the reference receiver R_(x) and a second secondary receiver R_(x2)for the hyperbola 44. The set of possible places of location of thetarget 31 then includes the points of intersection 46 and 47 of the twocurves. Consequently, to determine the position of the target 31 withoutambiguity it is necessary to have at least one additional secondaryreceiver (“2D multiple secondary receivers” configuration), theintersection of the three location hyperbola 41, 44 and 48 defining asingle common point, the point 46 in the example, on which the target 31is situated.

The theoretical determination principle explained in the precedingparagraphs through a two-dimensional location (“2D” location) cannaturally be extended to a location in space (i.e. in 3D location).

In TOA mode, the location ellipse in “single-transmitter” mode isreplaced by an ellipsoid in space, an ellipsoid which can be representedby the following equation:

$\begin{matrix}{{\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{b^{2}}} = 1} & \lbrack 3\rbrack\end{matrix}$with:a=R _(b)/2and b=√{square root over (a ² −L ²/4)}

Similarly, in TDOA mode, the location hyperbola in “single secondaryreceiver” mode is replaced by a hyperboloid in space, a hyperboloidwhich can be represented by:

$\begin{matrix}{{\frac{x^{2}}{c^{2}} - \frac{y^{2}}{d^{2}} - \frac{z^{2}}{d^{2}}} = 1} & \lbrack 4\rbrack\end{matrix}$with:c=R _(d1)/2and d=√{square root over (L ²/4−c ²)}.

Interest is now focused on FIGS. 5 and 6 which illustrate, throughsimple location scenarios in TOA 2D mode (FIG. 5) and in TDOA 2D mode(FIG. 6), the problem raised by the accuracy of the real measurementsobtained with the receivers.

Like any measuring device, the receivers used by the passive locationsystems provide measurements that are marred by a certain inaccuracythat can be assumed to be bounded. This inaccuracy means that thedistance measurements performed in TOA or TDOA mode are represented, nolonger by exact values, but by intervals, the size of which correspondsto the maximum measurement error. These intervals [x] are defined by thefollowing equation:[x]=[x ⁻ ,x ⁺ ]={x ε

/x ⁻ ≦x≦x ⁺}  [5]in which

represents the set of real numbers.

If σ_(R) _(b) represents the maximum error on the measurement R_(b) inTOA mode, a measurement of R_(b) with bounded error is defined by theinterval:[R _(b) ⁻ ,R _(b) ⁺ ]:=[R _(b)−σ_(R) _(b) ,R _(b)+σ_(R) _(b) ]  [6]

Consequently, taking into account a bounded error on the measurement ofR_(b) in “TOA 2D” mode leads to the target concerned being located noton a curve but on a location surface delimited by two confocal ellipses51 and 52 of respective parameters R_(b) ⁻ and R_(b) ⁺, such as thatillustrated in FIG. 5. This surface 53 can be described by the followingparametrical form:

$\begin{matrix}\left\{ {\left( {x,y} \right) \in \mathcal{R}^{2}} \middle| {1 \in {\frac{x^{2}}{\lbrack a\rbrack^{2}} + \frac{y^{2}}{\lbrack b\rbrack^{2}}}} \right\} & \lbrack 7\rbrack\end{matrix}$in which:[a]=[R _(b)]/2and [b]=√{square root over ([a] ² −L ²/4)}are evaluated by applying the interval computation rules.

Similarly, if a location is performed in space (“TOA 3D”), the takinginto account of the bounded errors of the measurement of R_(b) leads tothe location of the target concerned in a volume defined by two confocalellipsoids, a volume that can be described by the following parametricalform:

$\begin{matrix}\left\{ {\left( {x,y,z} \right) \in \mathcal{R}^{3}} \middle| {1 \in {\frac{x^{2}}{\lbrack a\rbrack^{2}} + \frac{y^{2}}{\lbrack b\rbrack^{2}} + \frac{z^{2}}{\lbrack b\rbrack^{2}}}} \right\} & \lbrack 8\rbrack\end{matrix}$

As for the TOA mode, the taking into account of a bounded error on themeasurement of R_(d1) in “TDOA 2D” mode leads to the location of thetarget concerned not on a curve but on a location surface delimited bytwo confocal hyperbolas 61 and 62 of respective parameters R_(d1) ⁻ andR_(d1) ⁺ defined by the interval [R_(d1) ⁻,R_(d1) ⁺]:=[R_(d1)−σ_(R)_(d1) ,R_(d1)+σ_(R) _(d1) ], such as that illustrated in FIG. 6. Thissurface 63 can be described by the following parametrical form:

$\begin{matrix}\left\{ {\left( {x,y,z} \right) \in \mathcal{R}^{2}} \middle| {1 \in {\frac{x^{2}}{\lbrack c\rbrack^{2}} - \frac{y^{2}}{\lbrack d\rbrack^{2}}}} \right\} & \lbrack 9\rbrack\end{matrix}$with:[c]=[R _(d1)]/2and [d]=√{square root over (L ²/4−[c] ²)}

In the case of a location in space (“TDOA 3D”), the taking into accountof the bounded errors on the measurement of R_(d1) leads to the locationof the target concerned in a volume defined by two confocalhyperboloids, a volume that can be described, in a similar manner, bythe following parametrical form:

$\begin{matrix}\left\{ {\left( {x,y,z} \right) \in \mathcal{R}^{3}} \middle| {1 \in {\frac{x^{2}}{\lbrack c\rbrack^{2}} - \frac{y^{2}}{\lbrack d\rbrack^{2}} - \frac{z^{2}}{\lbrack d\rbrack^{2}}}} \right\} & \lbrack 10\rbrack\end{matrix}$

Interest is now focused on FIG. 7 which schematically shows theprinciple of the method according to one or more embodiments of theinvention. In order to make the explanation of the operating principleof the method according to the invention clearer, this principle isdescribed here in detail for the particular case of the search for anarea of a location by means of location systems of “single-transmitter”type (location in TOA mode) or “single secondary receiver” type(location in TDOA mode).

The basic principle of the method involves a progressive refining of thelocation area. It includes subdividing the space that is to be analyzed(i.e., the initial search block [x₀]×[y₀] into adjoining subblocks inwhich the presence of a target is evaluated by means of an ad hoccriterion. The subblocks in which the presence of a target is confirmedare in turn subdivided and the others are rejected. The resultingiterative process is repeated as long as the presence of a target isconfirmed in the blocks currently being analyzed and a stop criterion(corresponding to a block width objective) is not reached.

To this end, the method according to one or more embodiments of theinvention includes a number of steps:

-   -   an initialization step 71,    -   an iterative calculation step 72.

The initialization step 71 includes defining an initial block, [X₀],corresponding to an a priori search space. In 2D mode, the followingthus applies:[X ₀ ]=[x ₀ ]×[y ₀], andand in 3D mode, the following applies:[X ₀ ]=[x ₀ ]×[y ₀ ]×[z ₀].

The step 71 also involves initializing a list L₀, of a size that variesduring the implementation of the method, including the list of blocks tobe studied. The content of L₀ is initialized with [X₀].

The step 71 then includes defining a stop criterion for the method. Thisstop criterion is given here by the “objective” resolution Δ^(obj) withwhich it is desired to ultimately characterize the location area. Thisresolution is naturally limited by the accuracy of the measurementssupplied by the receivers, but it can be arbitrarily set within thislimit. In 2D location mode, it is perhaps defined in a coordinate systemxOy by Δ^(obj)=(Δ_(x) ^(obj),Δ_(y) ^(obj)).

The step 71 is followed by a step 72 carrying out an iterativeprocessing operation that includes two nested processing loops, a mainloop 73 and a secondary loop 74.

The main loop 73 includes updating the list L₀ established oninitialization with the results of the processing carried out by thesecond loop. Thus, on each iteration i of the main loop, there is are-updated list L₀ available, denoted L^(i) ₀.

On each iteration, each block └X_(0,k) ^(i)┘=L^(i) ₀(k) forming the listL^(i) ₀ is subdivided into N adjoining subblocks (N=4 in 2D mode, N=8 in3D mode) and grouped together in a list L₁.

As for the secondary loop 74, this includes eliminating from L₁, theblocks that are incompatible with the location area being sought (thatis, the blocks that do not validate the criterion).

Consequently, the element └X_(0,k) ^(i)┘ of L^(i) ₀ is replaced, inL^(i) ₀, by the list L₁.

According to one or more embodiments of the invention, the method ischosen to determine whether a subblock [X^(j)] includes a portion of thelocation area includes determining whether one or more points of thesubblock belong to that area.

To do this in the “single-transmitter” or “single secondary receiver”systems, simply the quantity J([X^(j)]) is considered, which is definedaccording to the location mode concerned by:

$\begin{matrix}{{{- {in}}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}\text{:}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack} & \lbrack 11\rbrack \\{{{- {in}}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}\text{:}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}} = {\quad\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} + \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack}} & \lbrack 12\rbrack \\{{{- {in}}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}\text{:}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack} & \lbrack 13\rbrack \\{{{- {in}}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}\text{:}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}\;} - \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack} & \lbrack 14\rbrack\end{matrix}$with [X^(j)]=[x^(j)]×[y^(j)] for the “2D” modes,and [X^(j)]=[x^(j)]×[y^(j)]×[z^(j)] for the “3D” modes.

Therefore, to verify if at least one point of a block [X^(j)] belongs tothe location area, it is sufficient, according to one or moreembodiments of the invention, to check whether:0 εJ([X^(j)]).

The main loop is intrinsically an endless loop for which it is necessaryto determine a stop condition and an operation to test for theappearance of this stop condition. According to one or more embodimentsof the invention, this stop condition is initialized in the step 71 andrelates to the resolution Δ^(obj) with which it is desired to define thelocation area. Thus, the main loop 73 includes a test operation 75executed at the end of processing on each iteration i. This condition isdefined by the following equation:

$\begin{matrix}{\Delta^{L_{0}^{i}{(1)}} \leq \Delta^{obj} \equiv \left\{ \begin{matrix}{\Delta_{x}^{L_{0}^{i}{(1)}} \leq \Delta_{x}^{obj}} \\{and} \\{\Delta_{y}^{L_{0}^{i}{(1)}} \leq \Delta_{y}^{obj}}\end{matrix} \right.} & \lbrack 11\rbrack\end{matrix}$in which Δ^(L) ^(i) ⁰ ⁽¹⁾ represents the resolution of the first blockforming the list L^(i) ₀, each block of the list having an identicalresolution at this stage.

Thus, as long as the condition of the equation [11] is found to beverified, the iterative calculation step 72 is repeated. The iterationstops only when the desired resolution is reached.

Thus, when applied to a two-dimensional TOA or TDOA location processingoperation, and for an iterative subdivision of each block into P=4subblocks, the method according to one or more embodiments of theinvention can be described by the following sequence of actions:

-   1. Definition of the initial size of an analysis cell (block):    [X ₀ ]=[x ₀ ]×[y ₀].-   2. Initialization of the list L₀ of the analyzed blocks: L₀ ⁰.-   3. Definition of the stop criterion: Δ^(obj)=(Δ_(x) ^(obj),Δ_(y)    ^(obj)).-   4. /beginning of main loop/: (Formation of L₀ ^(i))-   5. For n varying from 1 to the size of the list L₀ ^(i-1)-   6. subdivision of each block n into P=4 adjoining subblocks:    [X _(n) ¹ ]=[x ⁻,(x ⁻ +x ⁺)/2]·[y ⁻,(y ⁻ +y ⁺)/2]    [X _(n) ² ]=[x ⁻,(x ⁻ +x ⁺)/2]·[(y ⁻ +y ⁺)/2,y ⁺]    [X _(n) ³]=[(x ⁻ +x ⁺)/2,x ⁺ ]·[y ⁻,(y ⁻ +y ⁺)/2]    [X _(n) ⁴]=[(x ⁻ +x ⁺)/2,x ⁻]·[(y ⁻ +y ⁺)/2,y ⁺]-   7. creation of the list L₁ ^(n)={[X_(n) ¹], [X_(n) ²], [X_(n) ³],    [X_(n) ⁴]}-   8. /beginning of secondary loop/-   9. For j varying from 1 to 4-   10. Elimination of the subblocks [X_(n) ^(j)], that do not    satisfying the selection criterion 0 ε J([X_(n) ^(j)]): Formation of    L₁ ^(n).-   11. if L₁ ^(n) is not empty: replacement, in L₀ ^(i-1), of the block    [X_(n)] by the list L₁ ^(n),    -   if L₁ ^(n) is empty: elimination of the block [X_(n)] from the        list L₀ ^(i-1)-   12. /end of secondary loop/-   13. /end of main loop/(Formation of L₀ ^(i))-   14. Calculation of the resolution of the first element of L₀ ^(i):    Δ^(L) ^(o) ^(i) ⁽¹⁾=(x _(L) ₀ _(i) ₍₁₎ ⁺ −x _(L) ₀ _(i) ₍₁₎ ⁻ ,y    _(L) ₀ _(i) ₍₁₎ ⁺ −y _(L) ₀ _(i) ₍₁₎ ⁻)-   15. if Δ^(L) ⁰ ^(i) ⁽¹⁾≦Δ^(obj): return to /start of main loop/    -   else: end of procedure-   16. /end of procedure/

The principle of implementation of the method according to one or moreembodiments of the invention can be advantageously illustrated by FIGS.8 and 9.

FIG. 8 illustrates the implementation of the method according to one ormore embodiments of the invention on a location in TOA mode, with a “2Dsingle-transmitter” type system.

As can be seen the figure, the implementation of the method according toone or more embodiments of the invention is physically embodied in abreakdown into blocks of the space that includes the location area 81.This subdivision forms a mesh of the space, a mesh in which each cell isanalyzed to determine whether or not it includes a portion of thelocation area. The duly formed mesh is refined iteratively. During thesuccessive iterations, certain cells 83 formed by the division of alarger cell 82 including a portion 84 of the location area no longerinclude such portions. According to one or more embodiments of theinvention, the process of refining these “empty” areas then ceases,which makes it possible advantageously to avoid continuing, for suchcells, an analysis that is sterile and costly in computation workloadand to concentrate the refining on the cells where it presents a benefitfor determining as accurately as possible the location area 81. Anirregular mesh is thus advantageously obtained, in which the tight meshis only centered on the location area.

FIG. 9 illustrates in the same way the implementation of the methodaccording to one or more embodiments of the invention on a location inTDOA mode, with a system of the “2D single secondary receiver” type. Theimplementation of the method according to one or more embodiments of theinvention is embodied in the same way by a subdivision into blocks ofthe space that includes the location area 91. Areas 93 are also obtainedfor which the refining is not carried out and areas 94 for which it iscontinued.

The method according to one or more embodiments of the invention,detailed in the preceding paragraphs for the particular case of systemsof “TOA single-transmitter” or “TDOA single secondary receiver” typescan obviously be applied to the more complex location systems of “TOAmultiple transmitter” or “TDOA multiple secondary receiver” types in 2Dor in 3D. These types of systems can be used, by considering theintersections of the different location areas, to refine the location ofthe target being sought by limiting the location area to theintersections of the different areas obtained. Regarding these modes,the method according to one or more embodiments of the invention, asdescribed in the foregoing, remains applicable in principle. Only thecriterion for selection of a subblock used in the secondary loop of thestep 72 of the iterative processing operation is modified. Thus, if thesystem includes a total number N of transmitters (location in “TOAmultiple transmitter” mode) or receivers (location in “TDOA multiplesecondary receiver” mode), a subblock [X] will be selected if thefollowing equation is verified:

$\begin{matrix}{A = {{\sum\limits_{j = 1}^{N}a_{j}} = N}} & \lbrack 12\rbrack\end{matrix}$in which each element a_(j) represents the result of the calculation ofa selection criterion relating to a pairing (receiver R_(x), transmitterT_(x) ^(i)) (TOA location mode) or (reference receiver R_(x), secondaryreceiver R_(x) ^(i)) (TDOA location mode).

The elements a_(j) are defined by the following equations:

$a_{1} = \left\{ {{\begin{matrix}{{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{1}\left( \lbrack X\rbrack \right)}} \\{{0\mspace{14mu}{else}}\mspace{110mu}}\end{matrix}\ldots a_{N}} = \left\{ \begin{matrix}{{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{N}\left( \lbrack X\rbrack \right)}} \\{{0\mspace{14mu}{else}}\mspace{115mu}}\end{matrix} \right.} \right.$and in which J_(j)([X]) represents, for each individual system (R_(x),T_(xj)) or (R_(x), R_(xj)), the quantity J([X]) defined previously ([11,12, 13, 14]).

The FIG. 10 illustrate the implementation of the method in the case of alocation system of “TOA two-transmitters” type. As can be seen in thefigure, the mesh produced using the method according to one or moreembodiments of the invention is once again an irregular mesh for whichthe tight cells, of greater resolution, are located on the locationareas 101 and 102 defined by each individual system, and morespecifically on the portions 103 and 104 of these areas that constituteintersections and that represents the possible location areas of thetarget.

Turning to FIG. 11, this illustrates in a similar manner theimplementation of the method in the case of a location system of the“TDOA two secondary receivers” type. As for the preceding case, it canbe seen that the mesh produced by the method according to one or moreembodiments of the invention includes tight cells only in the places ofthe areas 113 and 114 representing the intersections of the locationareas 111 and 112 defined by each individual system.

Since no system can be perfect, a certain tolerance can be accepted asto the number of basic criteria J_(n)([X]) that are simultaneouslysatisfied. This tolerance is reflected by an acceptance of an interval[X] if A≧M (with M≦N).

The present invention extends, by adapting the basic criteriaJ_(n)([X]):

-   -   to the problems of multiple TOA location: use of a family of        basic criteria combining the basic criteria associated with each        of the TOA problems (this situation occurs when trying to locate        a target by analyzing two receivers that are not co-located        operating in TOA mode).    -   to the problems of multiple-TDOA location: use of a family of        basic criteria combining the basic criteria associated with each        of the TDOA sub-problems.    -   to the problems of mixed TOA/TDOA location: use of a family of        basic criteria combining the basic criteria associated with each        of the TOA and TDOA problems.

1. A method of passively locating a target in TOA or TDOA mode comprising the steps of: subdividing into blocks a space in which a location area is situated; and iteratively performing on the blocks, until a predetermined criterion is obtained, the steps of: searching within each block for a presence of at least one point belonging to the location area; excluding from the blocks any blocks that do not contain a point that belongs to the location area; resubdividing into subblocks any blocks that contain a point that belongs to the location area; replacing the blocks by the subblocks, wherein each iteration increases an accuracy of defining the location area, and the method further comprises the steps of: forming a current selection from a first block that corresponds to an a priori search space; forming a current list from the first block; iteratively performing the steps of: subdividing each block of a current list into a first, second, third, and fourth adjoining subblocks; searching the first, second, third, and fourth subblocks for at least one point forming part of the location area, to produce searched subblocks; selecting the searched subblocks having at least one point forming part of the location area, to produce selected subblocks; updating the current list to obtain an updated list, by performing the steps of: deleting blocks that have no selected subblocks; and replacing blocks not deleted by the first, second, third, and fourth subblocks to form an updated current list, wherein: the iteratively performed steps comprise a test to compare a resolution of the blocks that constitute the updated current list; and the iteratively performed steps stop when the resolution is greater than a desired resolution.
 2. The method as claimed in claim 1, wherein in a TOA 2D single transmitter location mode, a subblock is selected if the subblock satisfies a criterion determined in accordance with the following relationship: ${{0 \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack},$ wherein: a=R_(b)/2; b=√{square root over (a²−L²/4)}; L is a length from a transmitter to a receiver; R_(b) is a sum of a length from the transmitter to the target, and a length from the target to the receiver; x^(j) and y^(j) are Cartesian coordinates of a subblock j, for a frame of reference centered on a middle of a line segment linking the source and the receiver; and [X^(j)] is a j-th subblock of block [X].
 3. The method as claimed in claim 1, wherein in a TDOA 2D single secondary receiver location mode, a subblock is selected if the subblock satisfies a criterion determined in accordance with the following relationship: ${{0 \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack},$ wherein: c=R_(d1)/2; d=√{square root over (L²/4−c² )}; L is a length from a first receiver to a second receiver; R_(d1) is a difference between a length from the transmitter to the second receiver, and a length from the transmitter to the first receiver; x^(j) and y^(j) are Cartesian coordinates of a subblock j, for a frame of reference centered on a middle of a line segment linking the first receiver and the second receiver; and [X^(j)] is a j-th subblock of block [X].
 4. The method as claimed in claim 1, wherein in a TOA 3D single transmitter location mode, a subblock is selected if the subblock satisfies a criterion determined in accordance with the following relationship: ${0 \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2\;}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} + \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack$ wherein: a=R_(b)/2; b =√{square root over (a² −L²/4)}; L is a length from a transmitter to a receiver; R_(b) is a sum of a length from the transmitter to the target, and a length from the target to the receiver; x^(j), y^(j) and z^(j) are Cartesian coordinates of a subblock j, for a frame of reference centered on a middle of a line segment linking the source and the receiver; and [X^(j)] is a j-th subblock of block [X].
 5. The method as claimed in claim 1, wherein in a TDOA 3D single secondary receiver location mode, a subblock is selected if a point of the subblock satisfies a criterion determined in accordance with the following relationship: ${{0 \in {{J\left( \left\lbrack X^{j} \right\rbrack \right)}\mspace{14mu}{with}\mspace{14mu}{J\left( \left\lbrack X^{j} \right\rbrack \right)}}} = \left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\left\lbrack d^{2} \right\rbrack} - 1} \right\rbrack},$ wherein: c=R_(d1)/2; d=√{square root over (L²/4−c²)}; L is a length from a first receiver to a second receiver; Ra_(d1) is a difference between a length from the transmitter to the second receiver, and a length from the transmitter to the first receiver; x^(j), y^(j) and z^(j) are Cartesian coordinates of a subblock j, for a frame of reference centered on a middle of a line segment linking the first receiver and the second receiver; and [X^(j)] is a j-th subblock of block [X].
 6. The method as claimed in claim 1, wherein in a TOA multiple transmitters location mode or in a TDOA multiple secondary receivers location mode, a subblock is selected if the subblock satisfies a criterion determined in accordance with the following relationship; $A = {{\sum\limits_{j = 1}^{N}a_{j}} = N}$ wherein: N is a number of transmitters or secondary receivers that constitute the TOA or TDOA location system concerned, respectively; and the terms a_(j) are determined in accordance with the following relationships: $a_{1} = \left\{ {{\begin{matrix} {{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{1}\left( \lbrack X\rbrack \right)}} \\ {{{0\mspace{14mu}{else}};}} \end{matrix}a_{2}} = \left\{ {{\begin{matrix} {{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{2}\left( \lbrack X\rbrack \right)}} \\ {{{0\mspace{14mu}{else}};}} \end{matrix}\ldots a_{N}} = \left\{ \begin{matrix} {{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{N}\left( \lbrack X\rbrack \right)}} \\ {{{0\mspace{14mu}{else}},}\mspace{110mu}} \end{matrix} \right.} \right.} \right.$ wherein: for each individual single-transmitter or single secondary receiver system, a quantity J_(j)([X]) is determined in accordance with the following relationships: ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}}};$ ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} + \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}}};$ ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}}};$ and ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}}},$ wherein: a=R_(b)/2; b=√{square root over (a²−L_(TOA) ²/4)}; c=R_(d1)/2; d=√{square root over (L_(TDOA) ²/4−c²)}; L_(TOA) is a length from a transmitter to a first receiver; L_(TDOA) is a length from a first receiver to a second receiver; R_(b) is a sum of a length from the transmitter to the target, and a length from the target to the receiver; R_(d1) is a difference between a length from the transmitter to the second receiver, and a length from the transmitter to the first receiver; x^(j), y^(j) and z^(j) are Cartesian coordinates of a subblock j, for a frame of reference centered on a middle of a line segment linking the source and the receiver in TOA mode, or for a frame of reference centered on a middle of a line segment linking the first receiver and the second receiver in TDOA mode, respectively; and [X^(j)] is a j-th subblock of block [X].
 7. The method as claimed in claim 1, wherein in a TOA multiple transmitters location mode or in TDOA multiple secondary receivers location mode, for a number N of transmitters or secondary receivers constituting the TOA or TDOA location system concerned, respectively, a subblock is selected if the subblock satisfies a criterion A determined in accordance with the following relationship: $A = {{\sum\limits_{j = 1}^{N}a_{j}} \geq M}$ wherein: M is a pre-determined number, less than N, of transmitters or secondary receivers constituting the TOA or TDOA location system concerned, respectively; and the terms a_(j) are determined in accordance with the following relationships: $a_{1} = \left\{ {{\begin{matrix} {{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{1}\left( \lbrack X\rbrack \right)}} \\ {{0\mspace{14mu}{else}};} \end{matrix}a_{2}} = \left\{ {{\begin{matrix} {{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{2}\left( \lbrack X\rbrack \right)}} \\ {{0\mspace{14mu}{else}};} \end{matrix}\ldots a_{N}} = \left\{ \begin{matrix} {{1\mspace{14mu}{if}\mspace{14mu} 0} \in {J_{N}\left( \lbrack X\rbrack \right)}} \\ {{0\mspace{14mu}{else}},} \end{matrix} \right.} \right.} \right.$ wherein: for each individual single transmitter or single secondary receiver system, a quantity J_(j)([X]) is determined in accordance with the following relationships: ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}}};$ ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack a\rbrack^{2}} + \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} + \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack b\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}}};$ ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 2D^{''}\mspace{14mu}{mode}}};$ and ${{J\left( \left\lbrack X^{j} \right\rbrack \right)} = {\left\lbrack {\frac{\left\lbrack x^{j} \right\rbrack^{2}}{\lbrack c\rbrack^{2}} - \frac{\left\lbrack y^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - \frac{\left\lbrack z^{j} \right\rbrack^{2}}{\lbrack d\rbrack^{2}} - 1} \right\rbrack\mspace{14mu}{in}\mspace{14mu}{\,^{``}{TDOA}}\mspace{14mu} 3D^{''}\mspace{14mu}{mode}}},$ wherein: a=R_(b)/2; b=√{square root over (a²−L_(TOA) ²/4)}; c=R_(d1)/2; d=√{square root over (L_(TDOA) ²/4−c²)}; L_(TOA) is a length from a transmitter to a first receiver; L_(TDOA) is a length from a first receiver to a second receiver; R_(b) is a sum of a length from the transmitter to the target, and a length from the target to the receiver; R_(d1) is a difference between a length from the transmitter to the second receiver, and a length from the transmitter to the first receiver; x^(j), y^(j) and z^(j) are Cartesian coordinates of a subblock j, for a frame of reference centered on a middle of a line segment linking the source and the receiver in TOA mode, or for a frame of reference centered on a middle of a line segment linking the first receiver and the second receiver in TDOA mode, respectively; and [X^(j)] is a j-th subblock of block [X].
 8. The method as claimed in claim 6, wherein the method is used for the passive location of the target by a global system comprising individual single transmitter and single secondary receiver systems and operating in a mixed TOA/TDOA mode.
 9. The method as claimed in claim 7, wherein the method is used for the passive location of the target by a global system comprising individual single transmitter and single secondary receiver systems and operating in a mixed TOA/TDOA mode. 